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  <h1 id="数学考研-概率论" class="content-subhead">数学考研-概率论</h1>
  <p>
    <span>1970-01-01</span>
    <span><span class="post-category post-category-math">Math</span></span>
    <span id="/public/article/数学考研-概率论.html" class="leancloud_visitors" style="display:none" data-flag-title="数学考研-概率论"></span>
  </p>
  <h1 id="_1">第三部分 概率论</h1>
<h2 id="1">第1讲 随机事件和概率</h2>
<h4 id="1_1">1、概率基本公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(A\overline B) =&\ P(A-B) \\[1ex]
=&\ P(A)-P(AB) \\[3ex]
P(A+B) =&\ P(A)+P(B)-P(AB) \\[3ex]
P(A+B+C) =&\ P(A)+P(B)+P(C) \\[1ex]
&-P(AB)-P(AC)-P(BC)+P(ABC) \\[1ex]
\end{split}\end{equation}
</script>
</p>
<h4 id="2">2、条件概率</h4>
<p>
<script type="math/tex; mode=display">
P(A|B) = \cfrac{P(AB)}{P(B)}
</script>
</p>
<h4 id="3-pab">3、求 <script type="math/tex"> P(AB) </script>
</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(AB) &= P(B)P(A|B) = P(A)P(B|A) \ &\leftarrow 条件概率 P(A|B) = \cfrac{P(AB)}{P(B)}\\[2ex]
&= P(A) + P(B) - P(A+B) \ &\leftarrow P(A+B) = P(A)+P(B)-P(AB)\\[1ex]
&= P(A) - P(A\overline B) \ &\leftarrow P(A\overline B) = P(A)-P(AB)\\[1ex]
\end{split}\end{equation}
</script>
</p>
<h4 id="4">4、全概率公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\sum_{i=1}^nP(A_i) &= 1 \\[1ex]
A_1, A_2, ..., A_i &构成一个完备事件，且都有正概率\\[1em]
P(B) &= \sum_{i=1}^nP(A_i)P(B|A_i)
\end{split}\end{equation}
</script>
</p>
<h4 id="5">5、贝叶斯公式</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(A_j|B) &= \cfrac{P(A_jB)}{P(B)} \\[1ex]
&= \cfrac{P(A_j)P(B|A_j)}{\sum_{i=1}^nP(A_i)P(B|A_i)} = \cfrac{条件概率}{全概率}
\end{split}\end{equation}
</script>
</p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学考研-概率论.assets/截屏2020-12-19 22.58.17.jpg" alt="截屏2020-12-19 22.58.17" style="zoom:33%;" /></p>
<p><img class="pure-img" src="https://zromyk.gitee.io/myblog-figurebed/post/数学考研-概率论.assets/截屏2020-12-19 23.09.39.jpg" alt="截屏2020-12-19 23.09.39" style="zoom:33%;" /></p>
<h4 id="6">6、超几何分布</h4>
<p>
<script type="math/tex; mode=display">
p(k) = \cfrac{C_D^k * C_{N-D}^{n-k}}{C_N^n}
</script>
</p>
<h4 id="_2">独立性</h4>
<p>两个事件的独立性：<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
P(AB)=P(A)P(B)
&⟺A与B相互独立 \\[1ex]
&⟺A与\overline B相互独立 \\[1ex]
&⟺\overline A与B相互独立 \\[1ex]
&⟺\overline A与\overline B相互独立
\end{split}\end{equation}
</script>
<br />
多个事件的独立性：</p>
<p>若 <script type="math/tex"> A、B、C、D </script> 相互独立，则 <script type="math/tex"> AB </script> 与 <script type="math/tex"> CD </script> 相互独立， <script type="math/tex"> A </script> 与 <script type="math/tex"> BC-D </script> 相互独立（不包含对方事件）</p>
<h2 id="2_1">第2讲 随机变量的函数分布</h2>
<p>定理：设随机变量具有概率密度 <script type="math/tex"> f_X(x),-\infty\lt x\lt\infty </script> ，又设函数 <script type="math/tex"> y = g(x) </script> 处处可导，且恒有 <script type="math/tex"> g'(x)\lt0 </script> 或 <script type="math/tex"> g'(x)\gt0 </script> 则 <script type="math/tex"> Y=g(X) </script> 是连续型随机变量，其概率密度为<br />
<script type="math/tex; mode=display">
f_Y(y)=
\begin{cases}
f_X\{h(y)\}*|h'(y)| &\alpha\lt y\lt\beta\\[2ex]
0 &其他
\end{cases}
</script>
<br />
其中 <script type="math/tex"> \alpha=min\{g(-\infty),g(\infty)\},\beta=max\{g(-\infty),g(\infty)\} </script> ， <script type="math/tex"> x=h(y) </script> 为 <script type="math/tex"> y=g(x) </script> 的反函数。</p>
<h2 id="6_1">第6讲 数字特征</h2>
<h4 id="1_2">1、数学期望（均值）</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
E(Y) &= E[g(X)] = \int_{-\infty}^{\infty}g(x)f(x)dx \\[1ex] 
E(Z) &= E[g(X,Y)] = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}g(x,y)f(x,y)dxdy
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  E(C) &= C \\[1ex]
 E(CX) &= CE(X) \\[1ex]
E(X+Y) &= E(X)+E(Y) \\[2ex]
若X和Y独立\ E(XY) &= E(X)E(Y) 
\end{split}\end{equation}
</script>
</p>
<h4 id="2_2">2、方差</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
D(X) &= Var(X) \\[1ex]
&= E\{[X-E(X)]^2\} \\[1ex]
&= \int_{-\infty}^{\infty}[X-E(X)]^2f(x)dx \\[1ex]
&= E(X^2) - E[2XE(x)] + E[E(X)^2] \\[1ex]
&= E(X^2) - E(X)^2
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
  D(C) &= 0 \\[1ex]
D(X+C) &= D(X) \\[1ex]
 D(CX) &= C^2D(X) \\[1ex]
D(AX+BY) &= A^2D(X)+B^2D(Y) + 2ABCov(X,Y)
\end{split}\end{equation}
</script>
</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th align="left">分布</th>
<th>名称</th>
<th>概率密度</th>
<th align="center">
<script type="math/tex"> E(x) </script>
</th>
<th align="center">
<script type="math/tex"> D(x) </script>
</th>
</tr>
</thead>
<tbody>
<tr>
<td align="left">
<script type="math/tex"> X\sim (1,p) </script>
</td>
<td>0-1分布</td>
<td></td>
<td align="center">
<script type="math/tex"> p </script>
</td>
<td align="center">
<script type="math/tex"> (1-p)p </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim b(n,p) </script>
</td>
<td>两点分布</td>
<td></td>
<td align="center">
<script type="math/tex"> np </script>
</td>
<td align="center">
<script type="math/tex"> n(1-p)p </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim \pi(\lambda) </script>
</td>
<td>泊松分布</td>
<td>
<script type="math/tex"> p{\{x=k\}}=\cfrac{\lambda^ke^{-\lambda}}{k!} </script>
</td>
<td align="center">
<script type="math/tex"> \lambda </script>
</td>
<td align="center">
<script type="math/tex"> \lambda </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim U(a,b) </script>
</td>
<td>均匀分布</td>
<td>
<script type="math/tex"> f(x)=\cfrac{1}{b-a} </script> ，注意（ <script type="math/tex"> a\le x\le b </script> ）</td>
<td align="center">
<script type="math/tex"> \cfrac{a+b}{2} </script>
</td>
<td align="center">
<script type="math/tex"> \cfrac{(b-a)^2}{12} </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim Exp(\lambda) </script>
</td>
<td>指数分布</td>
<td>
<script type="math/tex"> f(x)=\lambda e^{-\lambda x} </script> ，注意（ <script type="math/tex"> x\gt0 </script> ）</td>
<td align="center">
<script type="math/tex"> \cfrac{1}{\lambda} </script>
</td>
<td align="center">
<script type="math/tex"> \cfrac{1}{\lambda^2} </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex"> X\sim N(\mu,\sigma^2) </script>
</td>
<td>正态高斯分布</td>
<td>
<script type="math/tex"> f(x)=\cfrac{1}{\sqrt{2\pi}\sigma}e^{-\cfrac{(x-\mu)^2}{2\sigma^2}} </script>
</td>
<td align="center">
<script type="math/tex"> \mu </script>
</td>
<td align="center">
<script type="math/tex"> \sigma^2 </script>
</td>
</tr>
<tr>
<td align="left">
<script type="math/tex">X\sim \chi^2(n)</script>
</td>
<td></td>
<td></td>
<td align="center">
<script type="math/tex"> n </script>
</td>
<td align="center">
<script type="math/tex"> 2n </script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="3">3、边缘概率密度</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_X(x) = \int_{-\infty}^{\infty}f(x,y)dy \\[1ex]
f_Y(y) = \int_{-\infty}^{\infty}f(x,y)dx
\end{split}\end{equation}
</script>
</p>
<h4 id="4_1">4、协方差</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
Cov(X,Y) &= E\{(X-E(X))(Y-E(Y))\} \\[1ex]
         &= E(XY) - E(X)E(Y) \\[2em]
Cov(X,Y) &= \int_{-\infty}^{\infty}\int_{\infty}^{\infty}(x-E(X))(y-E(Y))f(x,y)dxdy \\[1ex]
         &= \int_{-\infty}^{\infty}\int_{\infty}^{\infty}xyf(x,y)dxdy - \int_{-\infty}^{\infty}xf_X(x)dx\int_{-\infty}^{\infty}yf_Y(y)dy
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
Cov(aX,bY) &= ab*Cov(X,Y) \\[1ex]
Cov(X_1+X_2,Y) &= Cov(X_1,Y) + Cov(X_2,Y)
\end{split}\end{equation}
</script>
</p>
<h4 id="5_1">5、相关系数</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
\rho_{XY} = Corr(X,Y) &= \cfrac{Cov(X,Y)}{\sqrt{D(X)}\sqrt{D(Y)}}
\end{split}\end{equation}
</script>
</p>
<p>性质<br />
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X和Y相互独立&\Rightarrow(\rho_{XY} = 0)X和Y不相关 \\
&\nLeftarrow
\end{split}\end{equation}
</script>
</p>
<h2 id="7">第7讲 两个随机变量的函数的分布</h2>
<h5 id="1-zxy">1、 <script type="math/tex"> Z=X+Y </script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_{X+Y}(z)&=\int_{-\infty}^{\infty}f(z-y,y)dy \\
f_{X+Y}(z)&=\int_{-\infty}^{\infty}f(x,z-x)dx
\end{split}\end{equation}
</script>
</p>
<h5 id="2-zcfracyxzxy">2、 <script type="math/tex"> Z=\cfrac{Y}{X}、Z=XY </script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
f_{\frac{Y}{X}}(z)&=\int_{-\infty}^{\infty}|x|f(x,xz)dx \\
         f_{XY}(z)&=\int_{-\infty}^{\infty}\cfrac{1}{|x|}f(x,\frac{z}{x})dx
\end{split}\end{equation}
</script>
</p>
<h5 id="3-zminxyzmaxxy">3、 <script type="math/tex"> Z=min\{X,Y\}、Z=max\{X,Y\} </script>
</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
          Z &= min\{X,Y\}\\[1ex]
P\{M\le z\} &= 1-P\{M\gt z\}\\
            &= 1-P\{X\gt z,Y\gt z\}\\[1em]
          Z &= max\{X,Y\}\\[1ex]
P\{M\le z\} &= P\{X\le z,Y\le z\}
\end{split}\end{equation}
</script>
</p>
<h2 id="8">第8讲 统计量及其分布</h2>
<h4 id="1_3">1、统计量</h4>
<h5 id="1_4">1）样本均值</h5>
<h5 id="2_3">2）样本方差</h5>
<p>
<script type="math/tex; mode=display">
S^2=\cfrac{1}{n-1}\sum_{i=1}^{n}(X_i-\overline X)^2
</script>
</p>
<h5 id="3_1">3）样本标准差</h5>
<h5 id="4-k">4）样本 <script type="math/tex"> k </script> 阶原点距</h5>
<p>
<script type="math/tex; mode=display">
A_k=\cfrac{1}{n}\sum_{i=1}^{n}X_i^k
</script>
</p>
<h5 id="5-k">5）样本 <script type="math/tex"> k </script> 阶中心距</h5>
<p>
<script type="math/tex; mode=display">
B_k=\cfrac{1}{n}\sum_{i=1}^{n}(X_i-\overline X)^k
</script>
</p>
<h4 id="2_4">2、统计量四大分布</h4>
<h5 id="1-chi2">1） <script type="math/tex"> \chi^2 </script> 分布</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X_1,X_x,...,X_n&\sim N(0,1) \\[1em]
\Rightarrow X = \sum_{i=1}^{n}X_i^2&\sim\chi^2(n) \\[1em]
E(X) &= n \\[1ex]
D(X) &= 2n
\end{split}\end{equation}
</script>
</p>
<h5 id="2-t">2） <script type="math/tex"> t </script> 分布</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X &\sim N(0,1) \\
Y &\sim\chi^2(n),\ X和Y独立 \\[1em]
\Rightarrow t = \cfrac{X}{\sqrt{\cfrac{Y}{n}}}&\sim t(n) \\[1em]
t_{1-\alpha}(n) &= -t_{\alpha}(n)
\end{split}\end{equation}
</script>
</p>
<h5 id="3-f">3） <script type="math/tex"> F </script> 分布</h5>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
X &\sim\chi^2(n_1) \\
Y &\sim\chi^2(n_2),\ X和Y独立 \\[1em]
\Rightarrow F = \cfrac{X/n_1}{Y/n_2}&\sim F(n_1,n_2) \\[1em]
\cfrac{1}{F} &\sim F(n_2,n_1) \\[1ex]
F_{1-\alpha}(n_1,n_2) &\sim \cfrac{1}{F_\alpha(n_2,n_1)}
\end{split}\end{equation}
</script>
</p>
<h2 id="9">第9讲 参数估计</h2>
<h4 id="1_5">1、矩估计</h4>
<ol>
<li>利用 <script type="math/tex"> \overline X=E(x) </script>
</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      E(x)&=\int_{-\infty}^{\infty}xf(x;\theta)dx\\[1ex]
      E(x)&=\overline X\\[2ex]
\hat\theta&=f(\overline X)
\end{split}\end{equation}
</script>
</p>
<ol start="2">
<li>利用 <script type="math/tex"> \cfrac{1}{n}\sum X_i^2=E(x^2) </script>
</li>
</ol>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
      E(x^2)&=\int_{-\infty}^{\infty}x^2f(x;\theta)dx\\[1ex]
      E(x^2)&=\sum_{i=1}^n X_i^2\\[2ex]
\hat\theta&=f(\sum_{i=1}^n X_i^2)
\end{split}\end{equation}
</script>
</p>
<h4 id="2_5">2、最大似然估计</h4>
<p>
<script type="math/tex; mode=display">
\begin{equation}\begin{split}
                           L(\theta)&=\prod_{i=1}^nf(t_i;\theta)\\[1ex]
                      \ln{L{\theta}}&=\ln{\prod_{i=1}^nf(t_i;\theta)}\\[1ex]
\cfrac{d\{\ln{L{\theta}\}}}{d\theta}&=\cfrac{d\{\ln{\prod_{i=1}^nf(t_i;\theta)}\}}{d\theta} = 0\\[1ex]
                          \hat\theta&=\theta=f(\sum X_i)
\end{split}\end{equation}
</script>
</p>
<p>(2015年数学一)、若 <script type="math/tex"> g(n,\theta)=\cfrac{d\{\ln{L{\theta}\}}}{d\theta} </script> 为不含有 <script type="math/tex"> X_i </script> 的式子，当 <script type="math/tex"> g(n,\theta) </script> 随 <script type="math/tex"> \theta </script> 增大而增大时， <script type="math/tex"> \hat\theta=min\{X_1,X_2,...,X_n\} </script>
</p>
<h4 id="3_2">3、区间估计</h4>
<p><strong>置信区间：</strong>设总体 <script type="math/tex"> X </script> 的分布函数 <script type="math/tex"> F(x;\theta) </script> 含有一个未知参数 <script type="math/tex"> \theta </script> ， <script type="math/tex"> \theta\in\Theta </script> （ <script type="math/tex"> \Theta </script> 是 <script type="math/tex"> \theta </script> 的可能取值范围），对于给定值 <script type="math/tex"> \alpha(0\lt\alpha\lt1) </script> ，若由来自 <script type="math/tex"> X </script> 的样本 <script type="math/tex"> X_1,X_2,...,X_n </script> 确定两个统计量 <script type="math/tex"> \underline\theta=\underline\theta(X_1,X_2,...X_n) </script> ， <script type="math/tex"> \overline\theta=\overline\theta(X_1,X_2,...X_n) </script> （ <script type="math/tex"> \underline\theta\lt\overline\theta </script> ），对于任意 <script type="math/tex"> \theta\in\Theta </script> ，满足<br />
<script type="math/tex; mode=display">
P\{\underline\theta(X_1,X_2,...X_n)\lt\theta\lt\overline\theta(X_1,X_2,...X_n\} \ge 1-\alpha
</script>
</p>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th>项目</th>
<th>称为</th>
</tr>
</thead>
<tbody>
<tr>
<td>
<script type="math/tex"> (\underline\theta,\overline\theta) </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的<strong>置信区间</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> \underline\theta </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的双侧置信区间的<strong>置信下限</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> \overline\theta </script>
</td>
<td>
<script type="math/tex"> \theta </script> 的置信水平为 <script type="math/tex"> 1-\alpha </script> 的双侧置信区间的<strong>置信上限</strong></td>
</tr>
<tr>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td><strong>置信水品</strong></td>
</tr>
</tbody>
</table></div>
<h4 id="4_2">4、正态总体均值与方差的区间估计</h4>
<h5 id="nmusigma2">单个总体 <script type="math/tex"> N(\mu,\sigma^2) </script>
</h5>
<div class="pure-table-scrollable"><table class="pure-table pure-table-horizontal">
<thead>
<tr>
<th></th>
<th>置信水平</th>
<th>置信区间</th>
<th>置信区间</th>
</tr>
</thead>
<tbody>
<tr>
<td>均值 <script type="math/tex"> \mu </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td>
<script type="math/tex"> \sigma^2 </script> 已知： <script type="math/tex"> (\overline X\pm\cfrac{\sigma}{\sqrt{n}}z_{\alpha/2}) </script>
</td>
<td>
<script type="math/tex"> \sigma^2 </script> 未知： <script type="math/tex"> (\overline X\pm\cfrac{S}{\sqrt{n}}t_{\alpha/2}(n-1)) </script>
</td>
</tr>
<tr>
<td>方差 <script type="math/tex"> \sigma^2 </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td></td>
<td>
<script type="math/tex"> \mu </script> 未知： <script type="math/tex"> (\cfrac{(n-1)S^2}{\chi_{\alpha/2}^2(n-1)},\cfrac{(n-1)S^2}{\chi_{1-\alpha/2}^2(n-1)}) </script>
</td>
</tr>
<tr>
<td>标准差 <script type="math/tex"> \sigma </script> 的置信区间</td>
<td>
<script type="math/tex"> 1-\alpha </script>
</td>
<td></td>
<td>
<script type="math/tex"> \mu </script> 未知： <script type="math/tex"> (\cfrac{\sqrt{n-1}S}{\sqrt{\chi_{\alpha/2}^2(n-1)}},\cfrac{\sqrt{n-1}S}{\sqrt{\chi_{1-\alpha/2}^2(n-1)}}) </script>
</td>
</tr>
</tbody>
</table></div>
<h4 id="5_2">5、估计量的评选标准</h4>
<h5 id="1_6">1）无偏性</h5>
<p>若估计量 <script type="math/tex"> \hat\theta=\hat\theta(X_1,X_2,...X_n) </script> 的数学期望 <script type="math/tex"> E(\hat\theta) </script> 存在，且对于任意 <script type="math/tex"> \theta\in\Theta </script> ，有<br />
<script type="math/tex; mode=display">
E(\hat\theta)=\theta
</script>
<br />
则称 <script type="math/tex"> \hat\theta </script> 是 <script type="math/tex"> \theta </script> 的<strong>无偏估计量。</strong></p>
<blockquote class="content-quote">
<p>例如： <script type="math/tex"> E(\overline X)=\mu,E(S^2)=\sigma^2 </script> ，则对于任意分布，样本均值 <script type="math/tex"> \overline X </script> 是总体均值 <script type="math/tex"> \mu </script> 的无偏估计，样本方差 <script type="math/tex"> S^2=\cfrac{1}{n-1}\sum_{i=1}^n(X_i-\overline X)^2 </script> 是总体方差 <script type="math/tex"> \sigma^2 </script> 的无偏估计。</p>
</blockquote>
<h5 id="2_6">2）有效性</h5>
<p>设 <script type="math/tex"> \hat\theta_1=\hat\theta_1(X_1,X_2,...X_n) </script> 与 <script type="math/tex"> \hat\theta_2=\hat\theta_2(X_1,X_2,...X_n) </script> 都是 <script type="math/tex"> \theta </script> 的<strong>无偏估计量</strong>，若对于任意 <script type="math/tex"> \theta\in\Theta </script> ，有<br />
<script type="math/tex; mode=display">
D(\hat\theta_1)\le D(\hat\theta_2)
</script>
<br />
且至少对于某一个 <script type="math/tex"> \theta </script> 上式等号不成立，则称 <script type="math/tex"> \hat\theta_1 </script> 较 <script type="math/tex"> \hat\theta_2 </script> 有效</p>
<h5 id="3_3">3）相合性</h5>
<p>设 <script type="math/tex"> \hat\theta(X_1,X_2,...X_n) </script> 为参数 <script type="math/tex"> \theta </script> 的估计量，若对于任意 <script type="math/tex"> \theta\in\Theta </script> ，当 <script type="math/tex"> n\to\infty </script> 时， <script type="math/tex"> \hat\theta(X_1,X_2,...X_n) </script> 依概率收敛于 <script type="math/tex"> \theta </script> ，则称 <script type="math/tex"> \hat\theta </script> 为 <script type="math/tex"> \theta </script> 的相合估计量。</p>
<p>即，对于任意 <script type="math/tex"> \theta\in\Theta </script> 都满足：对于任意 <script type="math/tex"> \epsilon\gt0 </script> ，有<br />
<script type="math/tex; mode=display">
\lim_{n\to\infty}P\{|\hat\theta-\theta|\lt\epsilon\}=1
</script>
</p>
<p>则称 <script type="math/tex"> \hat\theta </script> 为 <script type="math/tex"> \theta </script> 的相合估计量。</p>
<h2 id="10">第10讲 中心极限定理</h2>
<p>设随机变量 <script type="math/tex">X_1,X_2,...,X_n</script> 独立同分布，并且具有有限的数学期望和方差：<script type="math/tex">E(X_i)=μ</script>，<script type="math/tex">D(X_i)=σ^2(k=1,2....)</script>，则对任意<script type="math/tex">x</script>，分布函数：<br />
<script type="math/tex; mode=display">
F_n=P\{\cfrac{\sum^n_{i=1}X_i-n\mu}{\sqrt{n}\sigma}\le x\}
</script>
<br />
当 <script type="math/tex">n</script> 很大时，<script type="math/tex">Y_n=\cfrac{\sum^n_{i=1}X_i-n\mu}{\sqrt{n}\sigma}</script> 近似地服从标准正态分布 <script type="math/tex">N(0,1)</script>
</p>
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  <a class="pure-menu-link" style="padding:0.1em 0em 0.1em 1.75em;" href="#1_3">1、统计量</a>
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